The bug
Villagers that have been infected by a zombie can despawn, even if they have been traded with. This makes it very difficult to rescue villagers after they have been infected, as the player may not move further than 128 blocks away from the infected zombie villager to gather all supplies needed to cure it. This might be intended.
To reproduce
Set difficulty to hard
Spawn a villager
Trade with the villager
Spawn a zombie
Let the zombie infect and convert the villager
Run
/tp @s ~ ~255 ~
→ ❌ The zombie villager despawned
Linked issues
is duplicated by
Comments

Yeah I saw this in Mumbo's video as well. It's clearly a bug named mobs aren't supposed to despawn.
For named villagers being able to despawn after being infected, please see MC-88967.
Can confirm this for 19w42a as well.
And confirmed in 19w45b.

This was supposed to be fixed in 1.16, yet three of my infected villagers despawned anyway. I haven't gone further away thay 70 blocks. It might be because they were fresh villagers, neither of them ever traded with. Do I need to trade with them so I don't need to name them or is it still a bug?
(playing in 1.16.2)
Issue has returned in 1.16.2. I have the same issue as Martin
Yes, I think it's made to not have hundreds of mobs not despawning. The only zombie villager you haven't trade with and still doesn't despawn is the one from the igloo, I believe
Happens even with a nametag. This happened in Mumbo Jumbo's latest video : https://youtu.be/WLygMYnQu4g?t=447